Fractional trajectories: Decorrelation versus friction

• We study the connection between fractional calculus and subordination processes.
• A fractional trajectory is equivalent to the ensemble average trajectory resulting from subordination.
• The internal friction effects associated to fractional derivatives can alternatively be explained by decorrelation.
• We use subordination theory to study the relaxation toward the equilibrium state induced by the fractional derivatives.
• As an example we apply the theory to the fractional Lotka–Volterra ecological model.

The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation of a fractional trajectory, that being an average over an ensemble of stochastic trajectories. Heretofore what has been interpreted as intrinsic friction, a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. We apply the general theory developed herein to the Lotka–Volterra ecological model, providing new insight into the final equilibrium state. The relaxation time to achieve this state is also considered.